Epicyclic Helical Channel for Parametric-resonance Ionization Cooling (PIC)

1. m Muons, Inc. Epicyclic Helical Channel for Parametric-resonance Ionization Cooling (PIC) V.S. Morozov Old Dominion University V. Ivanov, R.P. Johnson, M. Neubauer Muons, Inc. A. Afanasev Hampton University and Muons, Inc. A. S. Bogacz, Y.S. Derbenev Thomas Jefferson National Accelerator Facility K. Yonehara Fermi National Accelerator Laboratory Muon Collider Design Workshop BNL, December 1-3, 2009

2. m Muons, Inc. PIC Concept • Parametric resonance induced in muon cooling channel • Muon beam naturally focused with period of free oscillations • Wedge-shaped absorber plates combined with energy-restoring RF cavities placed at focal points (assuming aberrations corrected) • Ionization cooling maintains constant angular spread • Parametric resonance causes strong beam size reduction • Emittance exchange at wedge absorbers produces longitudinal cooling • Resulting equilibrium transverse emittances are an order of magnitude smaller than in conventional ionization cooling Muon Collider Design Workshop BNL, December 1-3, 2009

3. m Absorber plates Parametric resonance lenses w Muons, Inc. PIC Schematic • Equilibrium angular spread and beam size at absorber • Equilibrium emittance (a factor of improvement) Muon Collider Design Workshop BNL, December 1-3, 2009

4. m Muons, Inc. PIC Requirements • Varying dispersion • small at absorbers to minimize energy straggling • non-zero at absorbers for emittance exchange • large between focal points for compensating chromatic and spherical aberrations • Correlated optics • normal-mode oscillations’ periods must be low-integer multiples of dispersion magnitude oscillation period 1,2 / D = 1 or 2 • Required features can be produced by epicyclic magnetic field configuration • solenoid with two superimposed different-period transverse helical fields • uniform smoothly-varying fringe-field-free configuration Muon Collider Design Workshop BNL, December 1-3, 2009

5. m Muons, Inc. Simplified PIC Model • Two transverse helical fields with wave numbers k1 and k2 • Equation of motion • Cyclotron wave number • Analytic solution under approximation kc = const (pz = const) Muon Collider Design Workshop BNL, December 1-3, 2009

6. m Muons, Inc. Oscillating Dispersion • Dispersion function containing two oscillating terms • Condition for dispersion to periodically return to zero • k1= k2= kc/2  k gives B1= 9B2 and elliptic orbit • Dispersion function Muon Collider Design Workshop BNL, December 1-3, 2009

7. m Muons, Inc. Oscillating Dispersion (Cont.) p→p+Δp absorbers aberration correction Muon Collider Design Workshop BNL, December 1-3, 2009

8. m Muons, Inc. EPIC Based on HCC •   pT /pz~ 1gives more complicated picture • Since B2<< B1 , consider secondary helix perturbation • Start with single-periodic Helical Cooling Channel (HCC) • exact analytic solution • well-studied in simulations • orbit stability conditions Muon Collider Design Workshop BNL, December 1-3, 2009

9. m Muons, Inc. HCC Stability Region & Normal-Mode Tunes • In usual HCC sgn(k) = sgn(kc),  > 0 • In EPIC channel sgn(k) = sgn(kc),  < 0 Muon Collider Design Workshop BNL, December 1-3, 2009

10. m Muons, Inc. Adiabatic Turn On of Secondary Helix Muon Collider Design Workshop BNL, December 1-3, 2009

11. m Muons, Inc. Effective Force Approach • Introduce effective “friction” force • total energy conserved while phase volume is not • analogous to cooling • all trajectories converge towards periodic orbit • aids in finding periodic orbit when analytic solution is not available Muon Collider Design Workshop BNL, December 1-3, 2009

12. m Muons, Inc. EPIC Dispersion in Realistic Fields G4BeamLine p = 250 ± 12.5 MeV/c low  for primary helix Matlab p = 100 ± 1 MeV/c  ~ 1 for primary helix Muon Collider Design Workshop BNL, December 1-3, 2009

13. m Muons, Inc. Possible Technical Implementations • Solenoid with direct superposition of required helical harmonics • Adopt procedure developed for HCC by Kashikhin et al. • circular current loops centered on elliptical helix (different from periodic orbit) Muon Collider Design Workshop BNL, December 1-3, 2009

14. m Muons, Inc. Incorporating RF in 2-period HS • Hydrogen-pressurized RF cavity • Be wedge included in the walls replaces grids used with HPRF • many of required technical solutions already exist (using ceramic ring to adjust frequency, small-diameter power feed through…) • or Tapered no-RF cooling blocks with RF sections in between • Transverse-longitudinal coupling is minimal Muon Collider Design Workshop BNL, December 1-3, 2009

15. m Muons, Inc. Recent Progress • Found magnetic field configuration giving required dispersion • uniform • smoothly-varying • fringe-field free • Showed preliminary demonstration of needed features • Developed technique to identify periodic orbit • straightforward to find normal-mode tunes numerically using single-period orbital transfer matrix • Came up with ideas for technical implementation • Have ideas how to incorporate RF • Started preliminary G4BeamLine simulations Muon Collider Design Workshop BNL, December 1-3, 2009

16. Work Plan Numerical studies of orbital motion in epicyclic (double-periodic) channel Use developed tools to determine stability region around periodic orbit Determine characteristic parameters such as normal-mode tunes Learn to control parameters of motion with quadrupole and higher-order magnetic fields components Identify configuration and strengths of sextupole and octupole field components for aberration corrections Demonstrate in full-scale G4BeamLine simulations compensation of chromatic and spherical aberrations cooling and emittance exchange with wedge absorbers Explore possible technical solutions Study implementation of RF cavities as part of PIC channel design taking into account scattering in pressurizing gas if needed Investigate space-charge limitations m Muons, Inc. Muon Collider Design Workshop BNL, December 1-3, 2009 16